Part III Taking Chances for Fun and Profit

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1 Part III Taking Chances for Fun and Profit Chapter 8 Are Your Curves Normal? Probability and Why it Counts What You Will Learn in Chapter 8 How probability relates to statistics Characteristics of the normal (i.e., bell-shaped) How to compute z scores How to interpret z scores How to interpret the area under the normal Why Probability? The Normal Curve (a.k.a. The Bell Curve) Basis for the normal Provides basis for understanding probability of a possible outcome Basis for determining the degree of confidence that an outcome is true Example: Are changes in student scores due to a particular intervention that took place or by chance alone? Visual representation of a distribution of scores Three characteristics Mean, median, and mode are equal to one another Perfectly symmetrical about the mean Tails are asymptotic (get closer to horizontal axis but never touch) The Normal Curve Hey, That s Not Normal In general, many events occur in the middle of a distribution with a few on each end. 1

2 More Normal Curve 101 For all normal distributions Almost 100% of scores will fit between 3 and +3 standard from the mean So distributions can be compared Between different points on the x-axis, a certain percentage of cases will occur The distance between The mean and +1 standard deviation +1 and +2 standard +2 and +3 standard +3 standard and above Contains 34.13% of all the 13.59% of all the 2.15% of all the 0.13% of all the And the scores that are included (if the mean = 100 and the standard deviation = 10) are From 100 to 110 From 110 to 120 From 120 to 130 Above 130 The distance between The mean and -1 standard deviation Contains 34.13% of all the And the scores that are included (if the mean = 100 and the standard deviation = 10) are From 90 to 100 A standard score that is the result of dividing the amount that a raw score differs from the mean of the distribution by the standard deviation. -1 and -2 standard -2 and -3 standard 13.59% of all the 2.15% of all the From 80 to 90 From 70 to 80 ( X X) z s -3 standard and below 0.13% of all the Below 70 What about those symbols? 2

3 What about those symbols? z the z score X the individual score X the mean of the distribution s the distribution standard deviation Scores below the mean are negative (left of the mean), and those above are positive (right of the mean) A z score is the number of standard from the mean z scores across different distributions are comparable Using Excel to Compute z Score What z Scores Represent The areas of the that are covered by different z scores also represent the probability of a certain score occurring. So try this one In a distribution with a mean of 100 and a standard deviation of 10, what is the probability that one score will be 110 or above? What z Scores Represent How to Figure Out the Probability In a distribution with a mean of 100 and a standard deviation of 10, what is the probability that one score will be 110 or above? z 1 84% of all scores fall below a z score of +1 (50% below the mean and 34% above the mean). 16% of all scores fall above a z score of +1 (100% - 84% = 16%) 3

4 How to Figure Out the Probability STANDARDIZE a Raw Score Finding the Area Between Two Scores In a distribution with a mean of 100 and a standard deviation of 10, what is the area between a score of 110 and 125? Compute the individual z scores for the two raw scores z z 2.5 Finding the Area Between Two Scores Find (using Table B.1) the area between the mean and the z score of +1 = 34.13% Find (using Table B.1) the area between the mean and the z score of +2.5 = 49.38% For the area between the two scores, subtract the smaller from the larger 49.38% 34.13% = 15.25% Using a Drawing to Determine the Area Between Two Scores Using Excel for Probability Computes the probability with a function Table B.1 is not required NORM.S.DIST(z, cumulative) z is the z score for which you want the probability cumulative is a logical value Almost always True 4

5 What z Scores Really Represent Knowing the probability that a z score will occur can help you determine how extreme a z score you can expect before determining that a factor other than chance produced the outcome Keep in mind z scores are typically reserved for populations. 5